[1/4] You’re in interstellar space, and your radio is stuck on one band.
Can you head for your destination (yellow +) without missing a transmission from home (white dot) due to Doppler shift?
Yes — if the initial angle, θ, between home and your destination is greater than 90°.
Can you head for your destination (yellow +) without missing a transmission from home (white dot) due to Doppler shift?
Yes — if the initial angle, θ, between home and your destination is greater than 90°.
[2/4] If θ>90°, you can always find a velocity such that the signal from home is unshifted (then turn the antenna to keep it aimed at home).
But if θ>90°, why does the yellow + appear in the same hemisphere?
Because relativistic aberration shifts it from its initial bearing.
But if θ>90°, why does the yellow + appear in the same hemisphere?
Because relativistic aberration shifts it from its initial bearing.
[3/4] The group of all Lorentz transformations that keep a given beacon unchanged by Doppler shift and unmoved in the sky is actually isomorphic to the group of all rotations and translations of the Euclidean plane!
[4/4] Stereographically project the sky onto the plane, then rotate/translate before projecting back to the sphere. The north pole, which maps to ∞ on the plane, will be unchanged.
These actions correspond to Lorentz transformations that fix a beacon’s position and frequency.
These actions correspond to Lorentz transformations that fix a beacon’s position and frequency.
